Self-attractive random polymers

We consider a repulsion-attraction model for a random polymer of finite length in Z(d). Its law is that of a finite simple random walk path in Zd receiving a penalty e-(2beta) for every self-intersection, and a reward e(gamma/d) for every pair of neighboring monomers. The nonnegative parameters 0 and y measure the strength of repellence and attraction, respectively. We show that for gamma > beta the attraction dominates the repulsion; that is, with high probability the polymer is contained in a finite box whose size is independent of the length of the polymer. For gamma < beta the behavior is different. We give a lower bound for the rate at which the polymer extends in space. Indeed, we show that the probability for the polymer consisting of n monomers to be contained in a cube of side length epsilonn(1/d) tends to zero as n tends to infinity. In dimension d = 1 we can carry out a finer analysis. Our main result is that for 0 < gamma less than or equal to beta - 1/2 log 2 the end-to-end distance of the polymer grows linearly and a central limit theorem holds. It remains open to determine the behavior for gamma is an element of (beta-1/2 log 2, beta].